Why can't I see the transformation gizmo in the 3D View?
I don't see the left side of your Blender file, but maybe this helps:

If you click T a tool bar should appear on the left side of your screen. Here you can select for example the move or rotate tool and then you should see the correct gizmo. If you move or rotate the object using hot keys (for example G to move objects), there's no gizmo even if you turned on gizmos.